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The accurate and efficient evaluation of Newtonian potentials over general twodimensional domains is important for the numerical solution of Poisson's equation and volume integral equations. In this paper, we present a simple and efficient high-order algorithm for computing the Newtonian potential over a planar domain discretized by an unstructured mesh. The algorithm is based on the use of Green's third identity for transforming the Newtonian potential into a collection of layer potentials over the boundaries of the mesh elements, which can be easily evaluated by the Helsing--Ojala method. One important component of our algorithm is the use of high-order (up to order 20) bivariate polynomial interpolation in the monomial basis, for which we provide extensive justification. The performance of our algorithm is illustrated through several numerical experiments. [ABSTRACT FROM AUTHOR]

We apply the Thom–Milnor theorem to obtain the upper bounds on the amount of isolated (1) critical points of a potential generated by several fixed point charges(Maxwell’s problem on point charges), (2) critical points of SINR, (3) critical points of a potential generated by several fixed Newtonian point masses augmented with a quadratic term, (4) central configurations in the n-body problem. In particular, we get an exponential bound for Maxwell’s problem and the polynomial bound for the case of an “even dimensional” potential in Maxwell’s problem. [ABSTRACT FROM AUTHOR]

In this paper, the purpose is to discuss the existence and uniqueness of local solutions for a class of singular compressible non-Newtonian flows in one-dimensional bounded interval. The first difficulty is that the equation itself is singular and the initial condition is allowed to be vacuumed. The other is that the viscosity term and Newtonian potential term are completely nonlinear. [ABSTRACT FROM AUTHOR]

We prove that any positive solution for the critical nonlinear Hartree equation \begin{equation*} -\operatorname {\bigtriangleup }{u}\left (x\right) -\int _{\mathbb {R}^6} \frac {\left |{u}\left (y\right)\right |^2 }{ \left |x-y\right |^4 }\mathrm {d}y \,{u}\left (x\right)=0,\quad \quad x\in \mathbb {R}^6, \end{equation*} is nondegenerate. Firstly, in terms of spherical harmonics, we show that the corresponding linear operator can be decomposed into a series of one dimensional linear operators. Secondly, by making use of the Perron-Frobenius property, we show that the kernel of each one dimensional linear operator is finite. Finally, we show that the kernel of the corresponding linear operator is the direct sum of the kernel of all one dimensional linear operators. [ABSTRACT FROM AUTHOR]

We analyze mathematically the acoustic imaging modality using bubbles as contrast agents. These bubbles are modeled by mass densities and bulk moduli enjoying contrasting scales. These contrasting scales allow them to resonate at certain incident frequencies. We consider two types of such contrasts. In the first one, the bubbles are light with small bulk modulus, as compared to the ones of the background, so that they generate the Minnaert resonance (corresponding to a local surface wave). In the second one, the bubbles have moderate mass density but still with small bulk modulus so that they generate a sequence of resonances (corresponding to local body waves).We propose to use as measurements the far-fields collected before and after injecting a bubble, set at a given location point in the target domain, generated at a band of incident frequencies and at a fixed single backscattering direction. Then, we scan the target domain with such bubbles and collect the corresponding far-fields. The goal is to reconstruct both the, variable, mass density and bulk modulus of the background in the target region.1.We show that, for each fixed used bubble, the contrasted far-fields reach their maximum value at, incident, frequencies close to the Minnaert resonance (or the body-wave resonances depending on the types of bubbles we use). Hence, we can reconstruct this resonance from our data. The explicit dependence of these resonances in terms of the background mass density of the background allows us to recover it, i.e. the mass density, in a straightforward way.2.In addition, this measured contrasted far-fields allow us to recover the total field at the location points of the bubbles (i.e. the total field in the absence of the bubbles). A numerical differentiation argument, for instance, allows us to recover the bulk modulus of the targeted region as well. [ABSTRACT FROM AUTHOR]

The gravitational theory including spinor particle is formulated with the suggestion of tetrad fields E α μ by Utiyama, Kibble and Weinberg. 1 – 3 We might take the metric g μ ν = E α μ E β ν η α β as a constraint, that is used the method of Lagrange multiplier. In the weak-field approximation we calculate the metric of a static Dirac spinor particle proton and obtain that the metric is a diagonal g μ μ = η μ μ − 2 G N M / r which coincides with the anticipative result of Newtonian Gravitational potential. [ABSTRACT FROM AUTHOR]

In this note I give a short overview about convexity properties of solutions to elliptic equations in convex domains and convex rings and show a result about the optimal concavity of the Newtonian potential of a bounded convex domain in ℝn , n ≥ 3, namely: if the Newtonian potential of a bounded domain is ”sufficiently concave”, then the domain is necessarily a ball. This result can be considered an unconventional overdetermined problem.This paper is based on a talk given by the author in Bologna at the ”Bruno Pini Mathematical Analysis Seminar”, which in turn was based on the paper P. Salani, A characterization of balls through optimal concavity for potential functions, Proc. AMS 143 (1) (2015), 173-183.

Recently there has been a growing interest in quantum gravity theories with more than four derivatives, including both their quantum and classical aspects. In this work we extend the recent results concerning the non-singularity of the modified Newtonian potential to the most relevant case in which the propagator has complex poles. The model we consider is Einstein–Hilbert action augmented by curvature-squared higher-derivative terms which contain polynomials on the d'Alembert operator. We show that the classical potential of these theories is a real quantity and it is regular at the origin despite the (complex or real) nature or the multiplicity of the massive poles. The expression for the potential is explicitly derived for some interesting particular cases. Finally, the issue of the mechanism behind the cancellation of the singularity is discussed; specifically we argue that the regularity of the potential can hold even if the number of massive tensor modes and scalar ones is not the same.

In this paper, we study some regularity results of solutions of the Poisson equation $\triangle u=f,$ in Musielak-Orlicz spaces.

Abstract In this paper we rigorously justify the propagation of chaos for the parabolic–elliptic Keller–Segel equation over bounded convex domains. The boundary condition under consideration is the no-flux condition. As intermediate steps, we establish the well-posedness of the associated stochastic equation as well as the well-posedness of the Keller–Segel equation for bounded weak solutions. [ABSTRACT FROM AUTHOR]

It is proved that the family of all pairwise products of regular harmonic functions on D and of the Newtonian potentials of points on the line L ⊂ Rn is complete in L2(D), where D is a bounded domain in Rn, n ≥ 3, such that D¯ ∩ L = ∅. This result is used in the proof of uniqueness theorems for the inverse acoustic sounding problem in R3. [ABSTRACT FROM AUTHOR]

There is a conjecture that renormalizable higher-derivative gravity has a finite classical potential at the origin. In this work, we show clearly that the scale-invariant gravity (SIG) satisfies the conjecture. This gravity produces the better-behaved 1 / k 4 UV behavior as needed for renormalizability. It turns out that the SIG has the linear classical potential of V ∝ r and it is a UV complete theory. [ABSTRACT FROM AUTHOR]

We consider the minimum energy problem on the unit sphere $\mathbb {S}^{d-1}$ in the Euclidean space $\mathbb {R}^{d}$ , d = 3, in the presence of an external field Q, where the charges are assumed to interact according to Newtonian potential 1/ r , with r denoting the Euclidean distance. We solve the problem by finding the support of the extremal measure, and obtaining an explicit expression for the density of the extremal measure. We then apply our results to an external field generated by a point charge of positive magnitude, placed at the North Pole of the sphere, and to a quadratic external field. [ABSTRACT FROM AUTHOR]

We give universal inequalities for the norm of certain potential operators in terms of the least non-zero Neumann eigenvalue. [ABSTRACT FROM AUTHOR]

Recently there has been a growing interest in quantum gravity theories with more than four derivatives, including both their quantum and classical aspects. In this work we extend the recent results concerning the non-singularity of the modified Newtonian potential to the most relevant case in which the propagator has complex poles. The model we consider is Einstein–Hilbert action augmented by curvature-squared higher-derivative terms which contain polynomials on the d'Alembert operator. We show that the classical potential of these theories is a real quantity and it is regular at the origin despite the (complex or real) nature or the multiplicity of the massive poles. The expression for the potential is explicitly derived for some interesting particular cases. Finally, the issue of the mechanism behind the cancellation of the singularity is discussed; specifically we argue that the regularity of the potential can hold even if the number of massive tensor modes and scalar ones is not the same. [ABSTRACT FROM AUTHOR]

We study an intrinsic model for collective behaviour on the hyperbolic space H n. We investigate the equilibria of the aggregation equation (or equivalently, the critical points of the associated interaction energy) for interaction potentials that include Newtonian repulsion. By using the method of moving planes, we establish the radial symmetry and the monotonicity of equilibria supported on geodesic balls of H n. We find several explicit forms of equilibria and show that one such equilibrium is a global energy minimizer. We also consider more general potentials and utilize a technique used for R n to establish the existence of compactly supported global minimizers. Numerical simulations are presented, suggesting that some of the equilibria studied here are global attractors. The key tool in our investigations is a family of isometries of H n that we have developed for this purpose. • Isometries defined by the Lorentz transform on the hyperbolic space. • Analytical results for interaction potentials that contain Newtonian repulsion. • The existence of compactly supported global minimizers on the hyperbolic space. [ABSTRACT FROM AUTHOR]

We consider the minimal energy problem on the unit sphere $${\mathbb {S}}^2$$ in the Euclidean space $${\mathbb {R}}^3$$ immersed in an external field Q, where the charges are assumed to interact via Newtonian potential 1/ r, r being the Euclidean distance. The problem is solved by finding the support of the extremal measure, and obtaining an explicit expression for the equilibrium density. We then apply our results to an external field generated by a point charge, and to a quadratic external field. [ABSTRACT FROM AUTHOR]

One of the most cited papers in Applied Mechanics is the work of Eshelby from 1957 who showed that a homogeneous isotropic ellipsoidal inhomogeneity embedded in an unbounded (in all directions) homogeneous isotropic host would feel uniform strains and stresses when uniform strains or tractions are applied in the far-field. Of specific importance is the uniformity of Eshelby's tensor $\mathbf{S}$ . Following Eshelby's seminal work, a vast literature has been generated using and developing Eshelby's result and ideas, leading to some beautiful mathematics and extremely useful results in a wide range of application areas. In 1961 Eshelby conjectured that for anisotropic materials only ellipsoidal inhomogeneities would lead to such uniform interior fields. Although much progress has been made since then, the quest to prove this conjecture is still not complete; numerous important problems remain open. Following a different approach to that considered by Eshelby, a closely related tensor $\mathbf{P}=\mathbf{S}\mathbf{D}^{0}$ arises, where $\mathbf{D}^{0}$ is the host medium compliance tensor. The tensor $\mathbf{P}$ is associated with Hill and is of course also uniform when ellipsoidal inhomogeneities are embedded in a homogeneous host phase. Two of the most fundamental and useful areas of applications of these tensors are in Newtonian potential problems such as heat conduction, electrostatics, etc. and in the vector problems of elastostatics. Knowledge of the Hill and Eshelby tensors permit a number of interesting aspects to be studied associated with inhomogeneity problems and more generally for inhomogeneous media. Micromechanical methods established mainly over the last half-century have enabled bounds on and predictions of the effective properties of composite media. In many cases such predictions can be explicitly written down in terms of the Hill tensor, or equivalently the Eshelby tensor and can be shown to provide excellent predictions in many cases. Of specific interest is that a number of important limits of the ellipsoidal inhomogeneity can be taken in order to be employed in predictions of the effective properties of, for example, layered media and fibre reinforced composites and also to the cases when voids and cracks are present. In the main, results for the Hill and Eshelby tensors are distributed over a wide range of articles and books, using different notation and terminology and so it is often difficult to extract the necessary information for the tensor that one requires. The case of an anisotropic host phase is also frequently non-trivial due to the requirement of the associated Green's tensor. Here this classical problem is revisited and a large number of results for problems that are felt to be of great utility in a wide range of disciplines are derived or recalled. A scaling argument leads to the derivation of the Eshelby tensor for potential problems where the host phase is at most orthotropic, without the requirement of using the anisotropic Green's function. The Concentration tensor $\boldsymbol{\mathcal{A}}$ linking interior fields to those imposed in the far-field is derived for a wide variety of problems. These results can therefore be used in the various micromechanical schemes. Directly related to the tensors of Eshelby and Hill is the so-called Moment tensor $\mathbf{M}$ . As well as arising in the literature on micromechanics, this tensor is important in the vast area of research associated with inverse problems and specifically with the problem of identifying an object inside some domain given the application of a specific set of boundary conditions. Due to its fundamental importance and direct link to the Eshelby and Hill tensors, here we state the connection between $\mathbf{M}, \mathbf{P}$ and $\mathbf{S}$ in an effort to ensure that the work is of use to as wide a community as possible. Both tensor and matrix formulations are considered and contrasted throughout. Appendices give various details that illustrate the implementation of both approaches. [ABSTRACT FROM AUTHOR]